Cutting hyperplane arrangements

We consider a collectionH ofn hyperplanes in Ed (where the dimensiond is fixed). An ε-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all Ed and such that the interior of any simplex is intersected by at mostεn hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices (which is asymptotically optimal). Forr≤n1−σ, where δ>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn)O(1)rd−1). In the plane we achieve a time boundO(nr) forr≤n1−δ, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm.For ann point setX⊆Ed and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ϒ R; R is a simplex}),In timeO(n(logn)O(1)rd−1+rO(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ε-nets for other range spaces usually encountered in computational geometry.These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.

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